Scaling of optimal-path-lengths distribution in complex networks

We study the distribution of optimal path lengths in random graphs with random weights associated with each link (“disorder”). With each link i we associate a weight τ_i=exp(ar_i), where r_i is a random number taken from a uniform distribution between 0 and 1, and the parameter a controls the strength of the disorder. We suggest, in an analogy with the average length of the optimal path, that the distribution of optimal path lengths has a universal form that is controlled by the expression (1∕p_c)(ℓ_∞∕a), where ℓ_∞ is the optimal path length in strong disorder (a→∞) and p_c is the percolation threshold. This relation is supported by numerical simulations for Erdős-Rényi and scale-free graphs. We explain this phenomenon by showing explicitly the transition between strong disorder and weak disorder at different length scales in a single network.